### AIMS Mathematics

2023, Issue 3: 5484-5501. doi: 10.3934/math.2023276
Research article Special Issues

# On the Caputo-Hadamard fractional IVP with variable order using the upper-lower solutions technique

• Received: 16 October 2022 Revised: 03 December 2022 Accepted: 07 December 2022 Published: 19 December 2022
• MSC : 34A12, 34A40, 47A10

• This paper studies the existence of solutions for Caputo-Hadamard fractional nonlinear differential equations of variable order (CHFDEVO). We obtain some needed conditions for this purpose by providing an auxiliary constant order system of the given CHFDEVO. In other words, with the help of piece-wise constant order functions on some continuous subintervals of a partition, we convert the main variable order initial value problem (IVP) to a constant order IVP of the Caputo-Hadamard differential equations. By calculating and obtaining equivalent solutions in the form of a Hadamard integral equation, our results are established with the help of the upper-lower-solutions method. Finally, a numerical example is presented to express the validity of our results.

Citation: Zoubida Bouazza, Sabit Souhila, Sina Etemad, Mohammed Said Souid, Ali Akgül, Shahram Rezapour, Manuel De la Sen. On the Caputo-Hadamard fractional IVP with variable order using the upper-lower solutions technique[J]. AIMS Mathematics, 2023, 8(3): 5484-5501. doi: 10.3934/math.2023276

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• This paper studies the existence of solutions for Caputo-Hadamard fractional nonlinear differential equations of variable order (CHFDEVO). We obtain some needed conditions for this purpose by providing an auxiliary constant order system of the given CHFDEVO. In other words, with the help of piece-wise constant order functions on some continuous subintervals of a partition, we convert the main variable order initial value problem (IVP) to a constant order IVP of the Caputo-Hadamard differential equations. By calculating and obtaining equivalent solutions in the form of a Hadamard integral equation, our results are established with the help of the upper-lower-solutions method. Finally, a numerical example is presented to express the validity of our results.

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